Probably what you need to do is at a command prompt (possibly being run as an administrator) is to run the "netstat" or "netstat -b" command to see what network connections are currently active while you are streaming a video. Ideally, the only thing you are doing is watching the video so it's less confusing.
That would at least show where the actual video is being streamed from. Actually, just now I ran the Resource Monitor in Windows 7 and looked at the Network Tab and selected Firefox. Added the bytes recv column and started playing a video. For this specific video, everything was coming from 173.194.24.179 (a Google IP) and had this tracert:
>tracert 173.194.24.179
Tracing route to 173.194.24.179 over a maximum of 30 hops
1 1 ms 1 ms 1 ms 192.168.1.1
2 6 ms 9 ms 7 ms 10.2.0.1
3 5 ms 9 ms 9 ms COX-68-12-10-90-static.coxinet.net [68.12.10.90]
4 15 ms 10 ms 7 ms COX-68-12-8-50-static.coxinet.net [68.12.8.50]
5 19 ms 9 ms 9 ms 68.1.5.135
6 12 ms 17 ms 11 ms 72.14.212.237
7 * * * Request timed out.
8 18 ms 16 ms 19 ms 72.14.233.64
9 12 ms 18 ms 19 ms 72.14.232.187
10 12 ms 9 ms 9 ms 173.194.24.179
Trace complete.
Doing a tracert to youtube.com was completely different after the 6th hop.